Constructive Approximation

, Volume 34, Issue 3, pp 371–391 | Cite as

Regularity of Tensor Product Approximations to Square Integrable Functions



We investigate first-order conditions for canonical and optimal subspace (Tucker format) tensor product approximations to square integrable functions. They reveal that the best approximation and all of its factors have the same smoothness as the approximated function itself. This is not obvious, since the approximation is performed in L2.


Tensor products Low-rank approximation Optimal subspace approximation Sobolev spaces 

Mathematics Subject Classification (2000)

15A69 41A46 49N60 


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  1. 1.
    Bauer, H.: Measure and Integration Theory. De Gruyter, Berlin/New York (2001) MATHCrossRefGoogle Scholar
  2. 2.
    Beylkin, G., Mohlenkamp, M.J., Pérez, F.: Approximating a wavefunction as an unconstrained sum of Slater determinants. J. Math. Phys. 49, 032107 (2008) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chinnamsetty, S.R., Espig, M., Flad, H.-J., Khoromskij, B.N., Hackbusch, W.: Tensor product approximation with optimal rank in quantum chemistry. J. Chem. Phys. 127, 084110 (2007) CrossRefGoogle Scholar
  4. 4.
    Greub, W.H.: Multilinear Algebra. Springer, Berlin (1967) MATHGoogle Scholar
  5. 5.
    Helgaker, T., Jørgensen, P., Olsen, J.: Molecular Electronic-Structure Theory. Wiley, New York (1999) Google Scholar
  6. 6.
    Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991) MATHCrossRefGoogle Scholar
  7. 7.
    Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Khoromskij, B.N.: Tensor-structured numerical methods in scientific computing: Survey on recent advances. Preprint 21/2010 MPI MiS Leipzig (2010) Google Scholar
  9. 9.
    Le Bris, C.: A general approach for multiconfiguration methods in quantum molecular chemistry. Ann. Inst. H. Poincaré 11, 441–484 (1994) MATHGoogle Scholar
  10. 10.
    Lewin, M.: Solutions of the multiconfiguration equations in quantum chemistry. Arch. Ration. Mech. Anal. 171, 83–114 (2004) MathSciNetCrossRefGoogle Scholar
  11. 11.
    De Silva, V., Lim, L.-H.: Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl. 30(3), 1084–1127 (2008) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Tyrtyshnikov, E.: Preservation of linear constraints in approximation of tensors. Numer. Math. Theory Methods Appl. 2(4), 421–426 (2009) MathSciNetMATHGoogle Scholar
  13. 13.
    Uschmajew, A.: Well-posedness of convex maximization problems on Stiefel manifolds and orthogonal tensor product approximations. Numer. Math. 115(2), 309–331 (2010) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Yosida, K.: Functional Analysis. Springer, Berlin (1974) MATHGoogle Scholar
  15. 15.
    Yserentant, H.: Regularity and Approximability of Electronic Wave Functions. Springer, Berlin/Heidelberg (2010) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany

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